1.03D - Absolute Inequalities | Lecture Note

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High School
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2024
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Absolute Value Functions ABSOLUTE VALUE FUNCTIONS 1.03D – SOLVING INEQUALITIES 𝑓 𝑥 ≥𝑎 𝑥−ℎ +𝑘 𝑓 𝑥 <𝑎 𝑥−ℎ +𝑘 𝑎>0 𝑎<0 Where would I shade? • Steps: 1. 2. 3. 4. 5. 6. Sketch! Isolate absolute value Split into two cases: Negative and Positive Solve for 𝑥 in both Validate Write in interval notation Solving Inequalities with Algebra • 𝑥+1 −3≤2 • See as two separate functions •𝑓 𝑥 = 𝑥+1 −3 •𝑔 𝑥 =2 • 𝑓 𝑥 ≤ 𝑔(𝑥) Ex 1: 𝑓 𝑥 = 𝑥+1 −3 𝑔 𝑥 =2 𝑓 𝑥 ≤ 𝑔(𝑥) 𝑥+1 −3≤2 Isolate the Absolute 𝑥+1 ≤5 𝑥 + 1 ≥ −5 𝑥 ≥ −6 𝑥+1≤5 𝑥≤4 ∴ 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏 𝒊𝒔 𝒙 ∈ [−𝟔, 𝟒] Ex 1: Algebra 𝑥 + 1 − 3 ≤ −3 Isolate the Absolute 𝑥+1 ≤0 𝑥+1≤0 𝑥 ≤ −1 𝑪𝒉𝒆𝒄𝒌 𝑮𝒓𝒂𝒑𝒉 Ex 2: Algebra 𝑓 𝑥 = 𝑥+1 −3 𝑔 𝑥 = −3 𝑥 + 1 − 3 ≤ −3 Isolate the Absolute 𝑥+1 ≤0 𝑥+1≤0 𝑥 ≤ −1 ∴ 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏 𝒊𝒔 𝒙 = −𝟏 Ex 2: Algebra 𝑥 + 1 − 3 ≤ −4 Isolate the Absolute 𝑥 + 1 ≤ −1 𝐼𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡𝑖𝑜𝑛 𝐷𝑜𝑒𝑠 𝑁𝑜𝑡 𝐸𝑥𝑖𝑠𝑡 𝑪𝒉𝒆𝒄𝒌 𝑮𝒓𝒂𝒑𝒉 Ex 3: Algebra 𝑓 𝑥 = 𝑥+1 −3 𝑔 𝑥 = −4 • 𝑥−1 −4> 3𝑥 − 2 •𝑓 𝑥 = 𝑥−1 −4 •𝑔 𝑥 = 3𝑥 − 2 • 𝑓 𝑥 > 𝑔(𝑥) Ex 4: −5 −5 𝑓 𝑥 = 𝑥−1 −4 3𝑥 𝑔 𝑥 =− −5 2 𝑓 𝑥 > 𝑔(𝑥) 3𝑥 𝑥−1 −4>− −5 2 Isolate the Absolute 3𝑥 𝑥−1 >− −1 2 3𝑥 𝑥−1< +1 2 3𝑥 𝑥−1>− −1 2 𝑥 − <2 2 𝑥 > −4 𝑥>0 Ex 4: 5𝑥 >0 2 ∴ 𝒔𝒐𝒍𝒖𝒕𝒊𝒐𝒏 𝒊𝒔 𝒙 ∈ ] − 𝟒, ∞[ The solution set of an inequality with an absolute value 1. May be an interval or the union of intervals 2. May be a single value (one solution) 3. May be empty (no solution) Inequalities •Visions 1 •P55: #10a,c,e,f,h •P59: #26 Homework